<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi mathvariant="script">P</mml:mi><mml:mi mathvariant="script">T</mml:mi></mml:mrow></mml:math> -symmetric <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mo>−</mml:mo><mml:mi>g</mml:mi><mml:msup><mml:mrow><mml:mi>φ</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math> theory

The scalar field theory with potential $V(\ensuremath{\varphi})=\frac{1}{2}{m}^{2}{\ensuremath{\varphi}}^{2}\ensuremath{-}\frac{1}{4}g{\ensuremath{\varphi}}^{4}$ ($g&gt;0$) is ill defined as a Hermitian but in non-Hermitian $\mathcal{P}\mathcal{T}$-symmetric framework it well defined, and has positive real energy spectrum for the case of spacetime dimension $D=1$. While methods used literature do not easily generalize to quantum theory, this paper path-integral representation $\ensuremath{-}g{\ensuremath{\varphi}}^{4}$ shown provide unified formulation general $D$. A new conjectural relation between Euclidean partition functions ${Z}^{\mathcal{P}\mathcal{T}}(g)$ ${Z}_{\text{Herm}}(\ensuremath{\lambda})$ $\ensuremath{\lambda}{\ensuremath{\varphi}}^{4}$ ($\ensuremath{\lambda}&gt;0$) proposed: $\mathrm{log}{Z}^{\mathcal{P}\mathcal{T}}(g)=\frac{1}{2}\mathrm{log}{Z}_{\text{Herm}}(\ensuremath{-}g+\mathrm{i}{0}^{+})+\frac{1}{2}\mathrm{log}{Z}_{\text{Herm}}(\ensuremath{-}g\ensuremath{-}\mathrm{i}{0}^{+})$. This ensures theory. closely related rigorously valid $D=0$. For $D=1$, using semiclassical evaluation ${Z}^{\mathcal{P}\mathcal{T}}(g)$, verified by comparing imaginary parts ground-state ${E}_{0}^{\mathcal{P}\mathcal{T}}(g)$ (before cancellation) ${E}_{0,\mathrm{Herm}}(\ensuremath{-}g\ifmmode\pm\else\textpm\fi{}\mathrm{i}{0}^{+})$.